how do changes in speed affect the perception of...

Speed changes and time perception 1

How do changes in speed affect the perception of duration?

William J. Matthews

University of Essex, UK

Running head: Speed changes and time perception

Keywords: time perception; internal clock; speed; acceleration; deceleration; temporal estimation;

temporal reproduction

Author note

Affiliation: William J. Matthews, Department of Psychology, University of Essex.

Acknowledgements: This work was supported by a grant from the University of Essex Research

Promotion Fund. I am grateful to Rebeccah Billing for helpful discussion and assistance with data

collection. I thank Simon Grondin, Neil Stewart, Mair Thomas, John Wearden, and two anonymous

reviewers for comments on earlier versions.

Correspondence address: William Matthews, Department of Psychology, University of Essex,

Colchester, CO4 3SQ, United Kingdom, E-mail: [email protected]


Abstract

Six experiments investigated how changes in stimulus speed influence subjective duration.

Participants saw rotating or translating shapes in three conditions: constant speed, accelerating

motion, and decelerating motion. The distance moved and average speed were the same in all three

conditions. In temporal judgment tasks, the constant-speed objects seemed to last longer than the

decelerating ones, which in turn seemed to last longer than the accelerating stimuli. In temporal

reproduction tasks, the difference between accelerating and decelerating stimuli disappeared;

furthermore, watching an accelerating shape lengthened the apparent duration of the subsequent

(static) display. These results (a) suggest that temporal judgment and reproduction can dissociate for

moving stimuli because the stimulus influences the apparent duration of the subsequent interval,

and (b) constrain theories of time perception, including those which emphasize memory storage,

those which emphasize the existence of a pacemaker-accumulator timing system, and those which

emphasize the division of attention between temporal and non-temporal information processing.


Time winding down: How do changes in speed affect the perception of duration?

Suppose that you are standing on the platform of a railway station. Whilst waiting for your

train you watch several others move past: One glides through the station at a steady pace; a second

pulls out of the platform opposite and gathers speed before it disappears around the corner; and a

third hurtles out of a tunnel and slows to a halt to pick up passengers. Assuming that all three trains

are visible for the same length of time, will these three events all seem to have equal duration? Or

will the nature of the movement affect your perception of time?

Several investigators have examined the effects of stimulus speed on time perception. The

most widely-repeated finding is that moving stimuli seem to last longer than static ones (e.g., S. W.

Brown, 1995; Kanai, Paffen, Hogendoorn, & Verstraten, 2006; Lhamon & Goldstone, 1975), and that

apparent duration increases with increasing stimulus speed (e.g., Beckmann & Young, 2009; S. W.

Brown, 1995; Kaneko & Murakami, 2009; Leiser, Stern, & Meyer, 1991). Some caution is needed

before accepting the generality of these conclusions because the results are context-dependent: a

stationary oddball shown in a train of moving stimuli will have longer subjective duration (Tse,

Intriligator, Rivest, & Cavanagh, 2004).

The effects of stimulus speed on subjective duration have implications for any model of time

perception. Theories of time perception can be grouped into two broad classes (Lejeune, 1998). The

first class assumes that duration is a by-product of non-temporal information processing; there is no

specialized “timer” for measuring duration. For example, Ornstein (1969) argued that perceived

duration depends upon the space needed to store the stimulus in memory, and others have stressed

that time perception depends on the number of changes taking place during an interval (e.g., Block

& Reed, 1978; Poynter, 1983). These storage- or change-based models have typically been applied to

retrospective time judgments – where the participant does not know in advance that they will be

asked for a time estimate – but they have also been related to the effects of stimulus motion on

time perception in prospective tasks, where the observer knows that a time judgment will be

required: More complex motion patterns are taken to lengthen apparent duration (Aubry,

Guillaume, Mogicato, Bergeret, & Celsis, 2008) and faster speeds are taken to entail greater

positional change and, accordingly, longer subjective time (S. W. Brown, 1995; Poynter, 1989).

The second class of models assumes a specialized mechanism for measuring time, based

upon some kind of accumulation or counting process (e.g., Gibbon, Church, & Meck, 1984;

Rammsayer & Ulrich, 2001; Treisman, 1963; Treisman, Faulkner, Naish, & Brogan, 1990). There is

usually taken to be an internal pacemaker or oscillator which emits pulses that are accumulated until

stimulus offset, at which point the number of counts in the accumulator provides the measure of


duration. A subset of these pacemaker models include the idea that attention must be divided

between temporal processing and non-temporal processing, such that directing attention away from

the timer reduces pulse accumulation and shortens subjective duration (e.g., Zakay & Block, 1997).

Stimulus speed has been argued to affect the pacemaker, with faster speeds producing higher

accumulation rates and longer subjective time (Kaneko & Murakami, 2009).

The current work builds on previous investigations of the effects of movement on time

perception by asking, for the first time: how do changes in speed affect the perception of duration?

That is, if a moving stimulus remains visible for a fixed time, and travels a fixed distance, does its

apparent duration depend on whether it accelerates, decelerates, or moves at a constant speed?

Existing models do not generate clear predictions regarding the effect of changes to stimulus speed

on subjective duration because the relationship between stimulus movement and the relevant

theoretical constructs is rather under-specified. For example, does storage size depend purely on

distance moved – in which case accelerating, decelerating, and constant-speed stimuli will have the

same subjective duration – or do changes in speed entail an additional complexity, such that

constant-speed stimuli seem shorter than accelerating and decelerating objects? Similarly,

pacemaker-accumulator models do not provide strong predictions as they depend on the assumed

relationship between stimulus speed and pacemaker rate.

The aim of the current experiments is therefore to establish the phenomenology of the

effects of changes in speed on subjective duration. These effects will not allow us to choose between

competing models of time perception, but they will provide an important constraint on how these

theoretical positions accommodate the effects of stimulus motion on subjective time.

Experiment 1

Experiment 1 compared the apparent durations of rotating shapes which accelerated,

decelerated, or moved with constant speed. Participants rated the apparent durations of moving

shapes on a nine-point scale ranging from 1 (very short) to 9 (very long) (Goldstone & Goldfarb,

1964). For a given physical duration, distance moved and mean speed were kept constant while the

rate of change in speed varied.

Two points should be clarified from the outset. The first is that average speed, acceleration

rate, and distance travelled are unavoidably confounded. If a stimulus moves with constant velocity

v for a length of time t, one can display an accelerating stimulus for the same duration and with the

same mean velocity so that the distance travelled is the same. If the duration is then increased, one

has to choose between keeping the mean velocity the same as before (in which case the rate of


acceleration decreases and the distance travelled increases), keeping the rate of acceleration

constant (in which case the distance travelled and mean velocity will increase), or changing the

starting velocity. In the current experiments, mean speed was held constant across conditions as this

seemed simplest. The intrinsic relationships between time, distance, and rate of acceleration mean

that it is difficult to interpret any interaction between type of movement (accelerating or constant

velocity) and stimulus duration.

The second point to note is that the present experiments are concerned with apparent

duration, not with the accuracy of judgment. Studies of the accuracy of time judgments typically

vary the duration of some comparison stimulus with respect to a standard and require a

discrimination judgment (e.g., Ulrich, Nitschke, & Rammsayer, 2006b). Such an approach will be

difficult here because varying the comparison duration will necessarily alter some other aspect of

the stimulus. The current emphasis is therefore upon how stimulus dynamics affect perceived

duration.

Method

Participants. Twenty four participants took part (8 male; mean age 20.4 years, SD = 4.9

years). Most received course credit; one volunteered.

Stimuli. The stimuli were six 2D shapes (two rectangles, two squares, two crosses of varying

colours). The shapes were presented against a white background and measured no more than 256 x

256 pixels (sizes ranged from approximately 1.5 x 1.7 degrees to 3.5 x 4.0 degrees visual angle).

Stimuli were displayed on a 19” CRT monitor (1280 x 1024 pixels refreshing at 85 Hz), viewed from

approximately 1 m through the glass window of a sound-attenuating chamber. Stimulus timing was

controlled in terms of screen refreshes using PsychoPy (Peirce, 2007). The shapes rotated around

their centre about an axis perpendicular to the screen. In the constant speed condition, rotation was

a fixed 225 degrees per second (3 degrees per refresh). In the accelerating condition, the shape

smoothly accelerated from stationary such that the total distance moved (and hence the mean

speed) was the same as in the constant speed condition. The motion dynamics of the decelerating

condition mirrored those of the accelerating condition: the shape smoothly slowed from the

maximum speed of the accelerating condition to stationary. Stimuli were presented for either 2 s or

3 s. The starting orientation of each shape was a randomly selected integer from 0-359 degrees.

Design and Procedure. The study employed a 2 (duration, 2 s vs. 3 s) by 2 (movement

direction, clockwise vs. anticlockwise) by 3 (motion: constant, accelerating, decelerating) within-

subjects design. Participants judged each shape once in each combination of conditions, giving 72


test trials. Trial order was randomized for each participant. Participants completed 10 practice trials

during which stimuli (shape, movement, duration, direction) were randomly selected. The sequence

of events on each trial was: a blank interval for 706-1412 ms, a blank fixation cross for 353 ms, a

blank interval for 529-882 ms, followed by the stimulus (the slightly odd timings result from a minor

coding error). After stimulus offset, participants rated its apparent duration on a scale ranging from

1 (very short) to 9 (very long). In all experiments, participants were instructed not to count or use a

watch to time the stimulus but to rely on their subjective impressions. Across all experiments, a

small number of trials were excluded from analysis because of timing errors (dropped frames) during

stimulus display.

Results and Discussion

The results are shown in Figure 1. An initial analysis indicated no effect of rotation direction,

which was therefore dropped from the analysis. A 2 (duration) x 3 (condition) repeated measures

ANVOA indicated that, as one would expect, 3 s stimuli were judged to last longer than 2 s ones, F(1,

23) = 126.6, p <. 001, 2

p = .85. More importantly, motion dynamics affects rated duration, F(2,46) =

45.6, p < .001, 2

p = .67; constant-speed shapes seemed to last longer than decelerating objects,

which in turn seemed to last longer than accelerating stimuli. All Bonferroni-corrected pairwise

comparisons were significant (all corrected ps < .007). Motion condition also interacted with

duration, F(2,46) = 11.72, p < .001, 2

p = .34. The interaction reflects the fact that the effect of

stimulus duration (2 s vs. 3 s) is greater for the constant velocity condition than for the accelerating

and decelerating conditions. This effect did not replicate in the subsequent experiments and, as

noted above, the interaction is difficult to interpret because the final/starting speeds in the

accelerating and decelerating conditions vary with the duration in order to keep the mean speed

constant. It is worth noting that, because mean speed was held constant across conditions, the

distance travelled was greater in the 3 s condition than in the 2 s condition, and participants may

have used this distance cue as part of the basis for their temporal judgments. However, for both

physical durations, the distances travelled and mean velocities were the same for the accelerating,

decelerating, and constant speed conditions; only the change in stimulus speed can underlie the

observed differences in perceived duration. (The same is true of the other experiments reported

below.)

Experiment 1 shows that changes in stimulus speed influence perceived duration, and

produced the surprising result that constant speed stimuli seem to last longer than either

accelerating or decelerating shapes. Experiment 2 sought to clarify two methodological points.


Firstly, Experiment 1 used a subjective rating scale which the participant can use as he or she wishes.

Additional research from my lab suggests that judgments using this kind of scale are no different

from when the categories correspond to specific objective durations and the task is to identify the

stimuli as accurately as possible (e.g., response 1 = 0.75 s), but it is worth checking that the results

are not due to the use of this category judgment task. Secondly, perceptual judgments of a given

stimulus are influenced by the other items present in the experiment. For example, Gomez and

Robertson (1979) found that the oft-repeated positive relationship between stimulus size and

apparent duration only holds when both small and large items are intermixed in an experimental

session. Experiment 1 used a within-subjects design; perhaps the accelerating and decelerating

shapes are more perceptually similar, such that the constant speed condition “stands out” and

correspondingly seems to have longer duration (Pariyadath & Eagleman, 2007; Tse et al., 2004;

Ulrich et al., 2006b). Experiment 2 used a between-subjects design and participants made absolute

judgments of duration on a millisecond scale.

Experiment 2

Method

Participants. A total of 60 participants were recruited (the number suggested by an a priori

power calculation) but technical problems meant that only 51 provided data (11 males, mean age

21.4 years, SD = 6.2 years; 11 had completed the previous experiment but were naive as to the

results). Participants received course credit or £3.

Stimuli. The stimuli and apparatus were similar to those in Experiment 1. Four shapes were

used (a filled square, an unfilled square, a rectangle, a cross,) and all were circumscribed to be no

greater than 256x256 pixels (1.7 x 1.9 to 3.5 x 4.0 degrees). In the constant velocity condition,

rotation was a fixed 2 degrees per refresh (170 degrees per second). In the accelerating condition

the shape smoothly accelerated from stationary such that the total distance moved (and hence the

mean velocity) was the same as in the constant velocity condition. In the decelerating condition the

shape smoothly slowed from the maximum velocity of the accelerating condition to stationary.

Stimuli were presented for five different durations: 1000, 1247, 1494, 1741, 1988 ms (respectively

equal to 85, 106, 127, 148, or 169 refreshes). The starting orientation of each shape was a randomly

selected integer from 0-359 degrees.

Design and Procedure. Seventeen participants were assigned to each of the constant speed,

accelerating, and decelerating conditions. Each participant completed forty test trials, one

presentation of each shape moving in each direction (clockwise/anticlockwise) for each of the 5


durations. Participants completed 4 practice trials during which stimuli (shape, movement, duration,

direction) were randomly selected for judgment.

The sequence of events on each trial was: blank interval for 1000-2000 ms, fixation cross for

500 ms, blank interval for 750-1250 ms, stimulus presentation for 1000-1988 ms, followed by a blank

interval during which the participant typed his or her judgment of the duration of the stimulus.

Participants typed their judgments in milliseconds. (They were reminded that 1000 ms = 1 s, so that,

for example, 1.25 seconds = 1250 ms). They were told that no stimulus would be shorter than 100

ms or longer than 3000 ms and that they should be as precise as they could but that they need not

agonize over every decision. Responses outside the range 100-3000 ms were excluded. Participants

could see the values they typed appear on the screen, and could clear the entries and retype their

judgments before pressing the Return key to register the response and progress to the next trial. A

handful of trials were excluded because of missing responses.


The results are shown on the left of Figure 2. Inspection suggests that longer-lasting stimuli

are judged to last longer, and judgments are affected by motion: constant speed stimuli have the

longest perceived duration, followed by decelerating shapes, with accelerating objects having the

briefest subjective duration. A 3 (condition: constant, accelerating, decelerating) x 5 (stimulus

duration) mixed ANOVA revealed, confirmed these impressions, with a main effect of duration,

F(3.32, 159.37) = 260.3, p < .001, 2

p = .84 (here and at certain points below, a Huynh-Feldt

correction was applied because of violations of sphericity), a main effect of condition, F(2,48) = 3.28,

p = .046, 2

p = .12, but no interaction, F(6.64, 159.37) = 1.67, p = .123, 2

p = .07. Bonferroni-corrected

pairwise comparisons indicating that the constant velocity and accelerating conditions differed (p =

.041); no other comparisons were statistically significant (ps >.5), but this is likely due to the low

power of a between-subjects experiment with only 17 participants per condition.

As an additional analysis, I examined the variability of magnitude estimates. The coefficients

of variation (standard deviation of judgment divided by mean judgment) are shown in the right

panel of Figure 2. Variability decreased as the duration increased, F(3.41, 163.55) = 7.54, p < .001,

2

p = .14, but neither movement nor the movement-duration interaction were significant, Fs < 1.

The variability of temporal judgments is important to models of timing (e.g., Gibbon et al., 1984;

Ulrich, Nitschke, & Rammsayer, 2006a) but the interpretation of variability results from verbal

estimation tasks is not always straightforward (see e.g., Wearden, Norton, Martin, & Montford-


Bebb, 2007). The coefficient of variation data are therefore included for completeness, but will not

be discussed further.

To summarize: Experiment 2 therefore replicates the results of Experiment 1. Shapes

rotating with constant speed seem to have longer duration than those which decelerate, which in

turn seem to have longer duration than those which accelerate. This is true even when observers

only see one type of movement in the course of the experiment, and when they are attempting to

estimate the objective duration of the stimulus as accurately as they can.

Experiments 1 and 2 used rotational motion, which may have a privileged status in time

perception because of its periodicity. Experiment 3a sought to replicate the foregoing pattern of

results using translational motion.

Experiment 3a

Method

Participants. Twenty students from the University of Essex participated in exchange for

course credit or £3 (18 female, mean age 19.1, SD = 1.0). Nine had participated in one or both of the

experiments using rotating stimuli.

Stimuli. The stimulus was a black square (128 x 128 pixels, approximately 1.9 degrees visual

angle) presented against a white background. In the constant velocity condition, the square moved

at a constant 5 pixels per refresh (approximately 6.6 degrees visual angle per second). The

accelerating and decelerating conditions were like those in Experiments 1 and 2 and involved

smooth changes in speed with the same mean speed as the constant velocity case. Stimuli were

presented for 588, 941, 1294, or 1647 ms (50, 80, 110, or 140 frames). The square was always

presented halfway up the screen; its horizontal starting position was randomly chosen on each trial

such that the movement never carried the centre of the square outside an invisible boundary 400

pixels either side of the centre of the screen.

Design and Procedure. The experiment employed a 3 (movement: constant velocity,

accelerating, decelerating) x 4 (stimulus duration: 588, 941, 1294, 1647 ms) within subject design.

Participants were told they would receive 10 practice trials (although a programming error meant

that 8 participants only received 1 practice trial), during which stimuli (shape, movement, duration,

direction) were randomly selected for judgment. They then completed 96 test trials, comprising 8

presentations of each of the 4 stimulus durations in each of the 3 conditions. Movement was left to

right for half of the trials in each condition and right to left in the other half. Participants were

invited to take a short break after 32 and 64 test trials.


The sequence of events on each trial was as Experiment 2. After stimulus offset the

participant rated the apparent duration of each stimulus on a 9-point scale anchored at 1 (very

short) and 9 (very long) in the same way as Experiment 1.


The results are shown in the left panel of Figure 3. A 2 x 3 within-subjects ANOVA indicated

that longer physical durations were rated as lasting longer, F(1.90, 36.13) = 369.03, p < .001, 2

p =

.95, and that constant velocity stimuli were perceived as lasting longer than decelerating stimuli,

which were perceived as lasting longer than accelerating stimuli, F(2,38) = 23.63, p < .001, 2

p = .554.

Post-hoc tests revealed that all movement conditions differed from one another (all Bonferroni-

corrected ps < .02). Duration and movement condition did not interact, F(6,114) = 2.15, p = .053, 2

p

= .10.

For translation motion, as for rotation, the apparent duration follows the order constant

speed > decelerating > accelerating. So far, all of the experiments have assessed participants’

judgments of the stimulus duration. An alternative approach is to require participants to reproduce

the duration of the moving stimulus (e.g., S. W. Brown, 1995; Kanai et al., 2006). Experiment 3b

replicated Experiment 3a but used a reproduction paradigm to assess apparent duration.

Experiment 3b

Method

Participants. Twenty five students from the University of Essex took part (20 female; mean

age 19.3 years, SD = 2.0 years); some received course credit, others were paid £4. None had

participated in Experiment 3a; 9 had completed one or both of the earlier experiments.

Stimuli, Design, and Procedure. The stimuli and design were identical to Experiment 3a,

except that after stimulus offset there was a blank interval during which the participant reproduced

the stimulus duration by holding down the left mouse button. Participants were instructed to wait

until the stimulus had disappeared before responding (they could respond at any time after stimulus

offset). Release of the mouse button triggered the next trial.

Results

Responses shorter than 100 ms or longer than 4000 ms were discarded as mistakes; trials on

which the latency from stimulus offset to response was less than 150 ms were also discarded as


anticipatory responses. (Across the whole experiment, fewer than 2.7% of trials were discarded

because of display errors, extreme or anticipatory responses.)

The results are shown in the centre of Figure 3. As one would expect, reproduced duration

increased with stimulus duration, F(1.61, 38.60) = 240.37, p < .001, 2

p = .91. There was also a main

effect of movement condition, F(2,48) = 26.98, p < .001, 2

p =.53, but no interaction, F(5.21, 124.94)

= 1.88, p = .099, 2

p = .07. Bonferroni-corrected pairwise comparisons revealed that the constant

velocity condition was significantly different from both the accelerating and decelerating conditions

(ps < .001) which did not differ (p = .438).

Experiment 3b replicated the finding that the constant velocity condition was perceived as

lasting longer than the accelerating and decelerating conditions, but did not replicate the difference

between these latter conditions. This is unlikely to have been due to statistical power: in Experiment

3a, the effect size (Cohen’s d) for the difference between acceleration and deceleration was 0.74, so

that the power to detect this effect in the current experiment (even after the Bonferroni-adjusting

the alpha levels to .017) was approximately 85% (Erdfelder, Faul, & Buchner, 1996). [This power

calculation is based on the effect size observed in Experiment 3a (see e.g., Greenwald, Gonzalez,

Harris, & Guthrie, 1996), although some authors have queried this approach (e.g., Miller, 2009) so

caution regarding the exact power estimate may be appropriate.]

This failure to replicate is surprising, and may be due to the fact that participants were free

to begin their reproduction at any point after stimulus offset. If stimulus motion affects the time at

which reproduction begins, this may in turn influence the reproduced duration as memory for the

target duration is likely to change over time (Wackermann & Ehm, 2006; Wearden & Ferrara, 1993).

In Experiment 3c, the beginning of the reproduction interval was controlled.

Experiment 3c

Method

Participants. Twenty students from the University of Essex took part (14 female; mean age

19.4 years, SD = 3.1 years) for course credit or £3. Four had participated in one or both of

Experiments 1 and 2; none had participated in Experiments 3a or 3b.

Stimuli, Design, and Procedure. The stimuli, design, and procedure were identical to

Experiment 3b except for a change in the way the stimulus duration was reproduced. Following

stimulus offset, the screen was blank for 756 ms (65 frames) after which the black square

reappeared in the centre of the display. Participants were instructed to tap the spacebar when the


second, stationary square had been on the screen for the same length of time as the moving square.

When the participant pressed the spacebar, the square disappeared and the next trial began.


As before, responses shorter than 100 ms or longer than 4000 ms were discarded as

mistakes. The results are shown in the right panel of Figure 3. As one would expect, reproduced

duration increased with stimulus duration, F(2.11, 40.17) = 359.32, p < .001, 2

p = .95. There was also

a main effect of movement condition, F(2,38) = 14.90, p < .001, 2

p =.44, but no interaction, F(6, 114)

= 1.09, p = .374, 2

p = .05. Bonferroni-corrected pairwise comparisons revealed that the temporal

reproductions from the constant speed condition were significantly longer than those from both the

accelerating (p < .001) and decelerating conditions (p = .001) which did not differ (p = 1.000). The

results therefore replicate those of Experiment 3b under conditions in which the onset of the

temporal reproduction is controlled. The power to detect a difference between the accelerating and

decelerating conditions of the size found in Experiment 3a was approximately 75%. The power to

detect an effect in one or both of Experiments 3b and 3c is therefore 1-(0.15*0.25) = 96%, so the

lack of an effect is unlikely to be due to low power.

One possible explanation for the difference between the results using judgment tasks and

those using reproduction tasks is that the different types of stimulus motion – acceleration,

deceleration, or constant speed – differentially affect the apparent duration of stimuli that come

after the moving object. Temporal reproduction requires the second interval (the reproduction) to

be matched to the first. If the passage of subjective time during that second interval depends upon

the nature of the original stimulus, there may be a dissociation between the reproduced duration

and the judgment of the original duration. Experiment 4 asks whether acceleration, deceleration,

and constant speed differentially affect the perception of stimuli that come after the moving shape.

Experiment 4

This experiment was similar to Experiments 3a-3c. A square translated across the screen

with either constant, accelerating, or decelerating velocity, and was then replaced by a static square.

The moving square had the same duration on every trial, and was not the subject of judgment. The

second square remained on-screen for a variable amount of time and participants were required to

rate its duration on a nine-point scale. Because it is the static square which is to be judged, the

conditions are referred to as post-constant, post-acceleration, and post-deceleration.


Method

Participants. Twenty seven participants (19 female; mean age 21.5 years, SD = 8.0 years)

took part for course credit or £3. Some had participated in previous experiments.

Stimuli, Design, and Procedure. Participants were tested in quiet cubicles, viewing the

screen from approximately 1 m. On each trial, a blue square moved across the screen for 1647 ms

(140 frames) with either constant, accelerating or decelerating velocity, exactly as in Experiments 3a-

3c. After a 756-ms (65 frame) blank interval, a static red square appeared for 588, 941, 1294, or 1647

ms (50, 80, 110, or 140 frames). Participants rated the duration of this red square on a nine-point

scale, in the same way as for Experiments 1 and 3a. Participants were told that they should study the

moving square while it was on the screen, but that they only needed to judge the static square.


The data are shown in the left panel of Figure 4; the right panel shows the results collapsed

across duration. Longer-lasting stimuli were judged to last longer, F(1.53, 39.75) = 270.03, p < .001,

2

p =.91. Judgments were also affected by movement condition, F(2,52) = 3.67, p = .032, 2

p = .12,

but there was no interaction, F < 1. Inspection of the means in the right panel of Figure 4 suggests

that the post-accelerating condition produced smaller ratings than the post-decelerating and post-

constant speed conditions, which did not differ much. Bonferroni-corrected pairwise comparisons

partially confirmed this impression: the post-accelerating condition was judged shorter than the

post-constant speed case, p = .018, but the difference between post-accelerating and post-

decelerating conditions did not survive Bonferroni correction (p = .101; the uncorrected p value is

significant). As one would expect based on Figure 4, the post-constant speed and post-decelerating

conditions did not differ (p = 1.00). The finding that stimulus motion affects the apparent duration of

the subsequent static display provides an explanation for the dissociation between time judgment

and time reproduction seen in Experiments 3a, 3b and 3c; this idea is discussed in detail below.

General Discussion

The results may be summarized as follows. Experiment 1 found that rotating shapes are

perceived to last longer when they move with constant speed than when they decelerate, which in

turn seem to last longer than when they accelerate. Experiment 2 showed that this was not due to

intra-experiment stimulus comparisons or the choice of judgment task: trying to estimate physical


duration in milliseconds when one has only encountered one type of motion in the experiment

produces the same pattern. The results generalize to translational motion (Experiment 3a) but when

a temporal reproduction task is used the difference between acceleration and deceleration

disappears – although both still seem shorter than constant speed motion (Experiments 3b and 3c).

Finally, acceleration seems to reduce the apparent duration of stimuli which follow the moving

shape (Experiment 4).

These results are surprising and provide an important constraint on models of time

perception. They pose two core questions. First, how are we to explain the differences in the

apparent duration of constant-speed, accelerating, and decelerating objects? And second, why do

the subjective durations indicated by the judgment tasks differ from those indicated by the

reproduction tasks? These questions will be considered in turn, beginning with the latter.

The dissociation between judged duration and reproduced duration

Many studies use temporal reproduction to assess perceived duration (e.g., S. W. Brown,

1995; Kanai et al., 2006; Macar, Grondin, & Casini, 1994; Woodrow, 1930). It is usually implicitly

assumed that reproduction will produce the same qualitative pattern of results as other tasks, such

as magnitude estimation. This was not found in the current experiments.

A tentative explanation is that the three types of movement differentially affected the

passage of subjective time after stimulus offset. The reproduction tasks involved matching the

duration of a static, post-movement display to the remembered duration of a dynamic object. In

Experiment 4, the post-movement display appeared shorter after watching an accelerating shape

than after a constant-speed or decelerating shape. That is, it will take longer for the post-movement

display to “feel as long” after watching the accelerating shape as after the constant-speed or

decelerating stimuli. We can therefore reconcile the estimation and reproduction results as follows.

After stimulus offset, the encoded duration of the moving shapes follows the order constant speed >

decelerating > accelerating, and participants’ category judgments or magnitude estimates reflect this

ordering. However, the movement also affects the apparent duration of the post-stimulus display

such that it takes longer to reach a given subjective duration after viewing an accelerating shape.

Thus, the temporal reproductions of the accelerating and decelerating shapes are very similar: the

accelerating shape “seemed shorter” but it also takes longer for the post-stimulus display to match

this encoded duration. This argument is speculative, but it is consistent with the pattern found in

Experiments 3a, 3b, 3c and 4.

The idea that apparent duration depends upon the preceding stimulus is consistent with the

general finding that perceptual (and non-perceptual) judgments are shaped by the previously judged


item (e.g., DeCarlo & Cross, 1990; Matthews & Stewart, 2009a; Matthews & Stewart, 2009b; Ward &

Lockhead, 1970), and with evidence that is more specific to time perception. For example, G.D.A.

Brown et al. (2005) found that judgments of durations assimilated towards the stimulus presented

on the previous trial. The precise mechanism underlying the effects of changes in stimulus speed on

the perceived duration of subsequent items is a topic for future investigation. Broadly, we can

distinguish two alternatives.

The first is that the effects result from the perceptual similarity between the successive

stimuli. Novel stimuli seem to last longer than repetitions (Noguchi & Kakigi, 2006; Pariyadath &

Eagleman, 2007, 2008; Tse et al., 2004), and it may be that the static displays which followed the

moving shapes in Experiments 3b, 3c, and 4 were more perceptually different from the constant

speed and decelerating stimuli than they were from the accelerating stimuli, explaining the shorter

apparent duration in the latter case. This suggestion is post-hoc, but it carries the prediction that the

effect of stimulus motion on the apparent duration of subsequent stimuli will depend on the

properties of those stimuli – their similarity to the moving shape.

The second possibility is that stimulus motion exerts a more general change on the passage

of subjective time, such that the consequences will be the same irrespective of the properties of the

subsequent stimulus. For example, changes in stimulus speed may affect the rate of an “internal

pacemaker”. Wearden and colleagues have found that playing a rapid train of clicks before stimulus

presentation increases its perceived duration, and this effect increases with increasing stimulus

length consistent with “speeding up” an internal clock (Penton-Voak, Edwards, Percival, & Wearden,

1996; Wearden, Edwards, Fakhri, & Percival, 1998; Wearden et al., 2007). The naive expectation

might be that an accelerating stimulus will produce a faster subsequent pacemaker rate (i.e., the

opposite of what was found here), but the situation is likely to be more complicated, depending on

the characteristic frequency of the pacemaker, the frequencies of the moving stimuli, and the

coupling between the two (see e.g., Treisman et al., 1990).

As a subsidiary exploration of this idea, I examined the delay between stimulus offset and

the point where participants began reproduction in Experiment 3b. This latency may be taken as

some indication of the rate of passage of subjective time, with longer latencies indicating slower

rates. The results are plotted in Figure 5 (data treatment was as for the main analysis reported

above, except that trials where the latency was longer than 2.5 s were excluded). A repeated

measures ANOVA indicated no effect of stimulus duration and no interaction between duration and

movement [F(2.35, 56.27) = 1.92, p = .149 and F(6,144) = 1.65, p = .139 respectively]. However, the

time that people waited before beginning the reproduction did depend on the nature of the stimulus

movement, F(1.51, 36.12) = 5.31, p = .016, 2

p =.18; people seem to wait longer in the accelerating


condition than in the constant speed and decelerating conditions (Bonferroni-corrected ps = .032

and .065) which were very similar (corrected p = 1.00). These results are therefore congruent with

the framework outlined above, in which the passage of subjective time (pacemaker rate) is slower

after viewing an accelerating shape – although of course other explanations of this pattern are also

possible.

The core message is that temporal reproduction and temporal judgment need not produce

identical results because the to-be-reproduced stimulus can influence the apparent duration of the

subsequent interval during which reproduction takes place. The details of how this occurs comprise

an important direction for future work. Additional problems with the use of temporal reproduction

have recently been highlighted by Droit-Volet (2010).

The effects of changes in speed on judgements of duration

The primary topic of the current paper is the effect of changes in stimulus speed on

apparent duration. Experiments 1, 2, and 3a found that stimuli moving with constant speed are

judged to last longer than those which decelerate, which in turn seem to last longer than those

which accelerate. This pattern is surprising – although Grassi and Darwin (2006) have reported a

potentially related finding with auditory stimuli. In what follows I discuss how the current results

relate to three broad accounts of time perception, focussing on general principles rather than

specific theoretical models.

Memory, mental content, and the segmentation of experience. It has been argued that

perceived duration depends on the “mental content” or complexity of a stimulus – the storage space

needed to hold the event in memory (Ornstein, 1969). It is difficult to explain the current results in

terms of stimulus complexity. The stimuli in all conditions moved the same distance over the same

objective time. If anything, the accelerating and decelerating conditions represent greater

complexity because they entail changes in stimulus speed as well as changes in stimulus position, yet

it was the constant speed condition which appears to last longest.

Relatedly, several authors have suggested that perceived time depends on the number of

changes during the interval, with more changes producing longer apparent durations (Block & Reed,

1978; Poynter, 1989). For moving stimuli, it has been argued that moving stimuli seem to last longer

than static ones because they entail more spatial change, and fast-moving stimuli seem to last longer

than slow-moving ones for the same reason (e.g., Brown, 1995). It is unclear how this extends to the

case where stimulus speed also changes. If apparent duration is solely dependent on changes in

position then constant-speed, accelerating, and decelerating shapes should appear equal. To the

extent that variations in speed represent an additional change, the accelerating and decelerating


conditions should seem longer than the constant speed case. Neither pattern was found here.

Finally, it might be possible to construct a change-based model in which the observer monitors the

changes in spatial location on a moment-by-moment basis and integrates them over time; this leads

to conclusions indistinguishable from those of the accumulator-based models discussed in the next

section.

Complexity- and change-based accounts of perceived duration do not offer a straightforward

explanation of the current data. More generally, it has been argued that such accounts are better-

suited to retrospective judgments – those in which the observer does not know in advance that a

duration estimate will be required – where responses are based on remembered duration (Block &

Zakay, 1997; Grondin, 2001).

Counting, clocks, and accumulation. Many models assume that time perception is based on

some kind of counting or accumulation process (e.g., Creelman, 1963; Gibbon et al., 1984; Killeen &

Weiss, 1987; Treisman, 1963; Treisman et al., 1990). There is usually assumed to be an internal

pacemaker or oscillator which emits counts or pulses; stimulus timing involves counting the number

of pulses which occur during the to-be-timed interval. There are numerous instantiations of this idea

but for now it sufficient to focus on the common principle that timing involves accumulation, and it

will be convenient to think of this accumulation as a time-continuous process. Manipulations which

alter perceived duration are typically argued to affect the rate of accumulation (Matthews, Stewart,

& Wearden, in press; Wearden et al., 1998; Wearden et al., 2007), but they might also influence the

latency with which timing is begun or ended (Matthews, in press).

How might an accumulator model accommodate the current results? An obvious approach is

to assume that the rate of the accumulation depends upon the stimulus speed (Kaneko & Murakami,

2009). Figure 6a schematically illustrates the changes in speed that took place over the course of

stimulus presentation in the current experiments. The question then is: how do we relate these

speeds to the accumulation rate in such a way that the total number of counts is in the order

constant speed > decelerating > accelerating?

Consider first the fact that the constant speed case has the longest perceived duration. This

rules out a linear relation between accumulation rate and speed: even if the rate at the beginning of

the stimulus was weighted more heavily than the end (or vice-versa), the constant-speed case will

be intermediate between the other two. A straightforward solution is to assume a non-linear

coupling between stimulus speed and accumulation rate. Figure 6b shows the results when

accumulation rate is a logarithmic function of speed; other non-linear functions would serve as well,

but there is some indication that a logarithmic function is a sensible choice. (In a recent study,

Kaneko and Murakami (2009) examined the perceived duration of Gabor patches with a moving


carrier and reported a logarithmic relationship between speed and subjective time.) In the figure,

the relationship between accumulation rate and stimulus speed was modelled by r = 100 +

100*ln(1+s), where r is the rate of accumulation, s is speed, and the extra constants were arbitrarily

chosen to ensure that the accumulation always has a reasonably high non-zero rate and to avoid

taking the logarithm of zero for stationary stimuli. The perceived duration is obtained by integrating

the curves shown in Figure 6b. As required, the constant-speed stimulus has the longest subjective

duration.

Under this scheme, the accelerating and decelerating stimuli will always have the same

apparent duration because they are symmetric in time (the latter is equivalent to the former played

backwards). The observed ordering decelerating > accelerating can be obtained if we assume that

the counts from the early part of the stimulus are weighted more heavily than those from the end –

for example, because the observer is more attentive at the start of the stimulus. An attentional

“gate” between pacemaker and accumulator which contracts over the course of stimulus

presentation provides a possible mechanism for this (see below). Figure 6c shows the results of

applying an exponentially-decreasing weighting function to the accumulation rates from Figure 6b.

Specifically, the weighting function was w = exp(-0.1t) where t is the time since stimulus onset.

Again, the functional form is arbitrary: other decreasing functions can produce the same effect.

Integrating the curves in Figure 6c gives the required result: constant speed produces the

longest subjective duration, followed by deceleration, with the accelerating shape having the

shortest apparent duration. A different choice of parameters will produce different results, but the

argument shows that the surprising pattern of results from the current experiments can be obtained

by two reasonable assumptions – a non-linear relationship between stimulus speed and

accumulation rate, and a differential weighting of information accumulated during early and late

portions of the stimulus. Perhaps more importantly, any attempt to explain the current data in terms

of an instantaneous coupling between accumulation rate and stimulus speed must incorporate both

of these assumptions. A monotonic change in the weighting of pacemaker output over the course of

the stimulus cannot by itself make the constant speed condition longest because the speed of the

constant stimulus is always below that of one of the other two stimulus types (except at the middle

of the presentation, when all three stimuli have equal speed). Similarly, an arbitrary relationship

between pacemaker rate and stimulus speed cannot separate the accelerating and decelerating

conditions because they are symmetric in time.

I have suggested that the results arise from a non-linear positive relationship between

stimulus speed and accumulation rate, and a diminishing weighting of the pacemaker output over

the course of the stimulus presentation. However, exactly the same pattern of results can be


obtained by reversing these assumptions – that is, by positing that accumulation rate is inversely

related to stimulus speed, with a weighting function that emphasizes the most recent portions of the

stimulus. Greater weighting of more recent experience is plausible and might arise if, for example,

there is passive leakage of accumulated counts (cf. Wackermann & Ehm, 2006). The idea that the

rate of accumulation is greater for slower-moving stimuli is perhaps counter-intuitive, and runs

against the typical assertion that faster stimuli seem to last longer (Beckmann & Young, 2009; S. W.

Brown, 1995; Kaneko & Murakami, 2009). However, there is some evidence that the relationship

between speed and perceived duration is not uniformly positive (Predebon, 2002). Moreover, the

fact that accelerating shapes (which end with rapid motion) make the subsequent static display

appear shorter (Experiment 4) is consistent with a negative relationship between speed and

pacemaker rate – although, as noted above, there are other explanations for this result. In order to

clarify the relationship between pacemaker rate and stimulus speed, it would be helpful to compare

the subjective durations of stimuli moving with different mean speeds over a range of durations for

all three of the movement conditions studied here – constant speed, accelerating, and decelerating.

It has been assumed here that the coupling between stimulus speed and accumulation rate

is instantaneous. More subtle versions are also possible, in which there is a lag between changes in

stimulus speed and changes in accumulation rate. In addition, the foregoing assumes that the

observer is able to estimate stimulus speed independently of the accumulation rate – for example,

with relatively peripheral representations of speed feeding in to a more central pacemaker.

However, it is also possible that the pacemaker is used when computing stimulus speed, with the

observer determining how much the spatial position has changed between pulses and adjusting the

pulse-rate contingent upon the computed velocity. The current data do not allow us to test these

more complex formulations.

Attention and temporal processing. A related approach to time perception emphasizes the

importance of attention. It is often argued that observers may attend to either the temporal or non-

temporal aspects of a stimulus. Such accounts posit a limited attentional capacity which is shared

between a timer (responsible for time processing) and a stimulus-processor (responsible for

processing other stimulus features)(e.g., Thomas & Weaver, 1975). The temporal processor

accumulates “subjective time units” or pulses from a pacemaker (Zakay, 1989), and subjective

duration is related to the attention given to time: greater temporal processing entails greater accrual

of subjective time units and, in turn, longer apparent duration. Conversely, directing attention away

from time – for example, by requiring judgments about some non-temporal aspect of the stimulus –

shortens apparent duration (Casini & Macar, 1997; Macar et al., 1994).


This class of model has much in common with those of the previous section. Indeed, Zakay

and Block (1997) have modified the internal-clock model of Gibbon et al. (1984) to include an

attentional gate which controls the flow of pulses from the pacemaker to the accumulator (see

Lejeune, 1998, for a discussion). However, the attentional models incorporate an additional

mechanism by which stimulus features or task demands may influence time perception: perceived

duration may change because of a shift in the attention paid to non-temporal properties of the

stimulus, rather than because of a change in the rate of the pacemaker.

Attentional accounts provide an alternative explanation for the finding that constant-speed

stimuli seemed to have longer duration than accelerating or decelerating shapes. Suppose that

attention can either be directed towards timing the moving objects or processing their other

properties – including the pattern of their motion. For an object moving at constant speed, the

motion presumably becomes predictable and uninteresting relatively quickly, freeing attentional

resources to process the duration of the stimulus. When the stimulus speed continuously changes

during viewing, more attention is devoted to processing its non-durational features over the whole

course of presentation, entailing a shorter subjective duration. Additional assumptions would then

be needed to accommodate the difference between accelerating and decelerating stimuli.

Conclusion

Changes in stimulus speed affect judged and reproduced duration – although not in the

same way. The difference between the category rating/magnitude estimation tasks and the

reproduction results is likely due to a change in the subjective passage of time during the

reproduction interval; the basis for this will be an important direction for the future. It is too early to

offer a definitive explanation of the effects of speed changes on judged duration. However, the

results are not obviously accommodated by memory storage-based accounts of time perception,

and if they are to be explained by pacemaker-accumulator models then a number of assumptions

must be made about the relationship between stimulus dynamics and the rate of accumulation. The

idea that changes in stimulus speed capture attention and prevent temporal processing provides an

alternative perspective on the data, but additional assumptions will be needed in order to capture

the full pattern of results.

The current work has focussed on simple comparisons between accelerating, decelerating,

and constant-speed stimuli. A full understanding of how stimulus speed affects time perception will

require an examination of additional parameters, including the starting, terminal, and mean speeds;

the nature of the motion (e.g., linear vs. non-linear); the rate of acceleration/deceleration; and more

complex changes in object speed over the course of stimulus presentation.


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Figure captions

Figure 1. Results of Experiment 1. Error bars show standard errors. Heterogeneity of the error terms

meant that it was not possible to use a pooled error term, so the error bars were calculated

separately for each data point. For within-subjects and mixed designs these bars provide no

indication of the significance of differences between conditions (Masson & Loftus, 2003).

Figure 2. Results of Experiment 2. The key results are shown in the left panel, which plots the mean

magnitude estimates. The right panel shows the coefficients of variation. Error bars show standard

errors calculated separately for each data point and therefore provide no indication of the

significance of differences between conditions.

Figure 3. Results of Experiments 3a (left panel), 3b (centre panel), and 3c (right panel). Experiment

3a required category judgments; Experiments 3b and 3c required temporal reproduction. Error bars

show plus/minus one SEM, calculated separately for each data point.

Figure 4. Results of Experiment 4. The left panel shows the results organized by duration. Error bars

show plus/minus one SEM calculated separately for each data point. The right panel shows the

effects of stimulus motion, collapsed across duration. Here it was possible to obtain a pooled error-

term; the error bars show plus/minus 95% confidence intervals calculated for a within-subjects

design.

Figure 5. Mean latency to begin reproduction in Experiment 3b, collapsed across stimulus duration.

Error bars show plus/minus one SEM calculated separately for each data point (violations of

sphericity prevented the use of a pooled term) so the bars are rather large and provide no indication

of the significance of differences between means.

Figure 6. Schematic illustration of changes in stimulus speed and hypothetical changes in

accumulation rate. The left panel shows stimulus speed over the course of presentation for each

condition. The centre panel shows the corresponding accumulation rates, assuming a logarithmic

dependency on stimulus speed. The right panel shows the effects of assigning progressively less

weight to the pacemaker output as the stimulus proceeds.


Figure 1.


Figure 2.


Figure 3.


Figure 4.


Figure 5.


Figure 6.

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